mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
K : ℝ≥0
f : X → Y
hf : AntilipschitzWith K f
s : Set Y
this✝³ : MeasurableSpace X := borel X
this✝² : BorelSpace X
this✝¹ : MeasurableSpace Y := borel Y
this✝ : BorelSpace Y
⊢ dimH (f ⁻¹' s) ≤ dimH s

Tactic:
refine dimH_le fun d hd => le_dimH_of_hausdorffMeasure_eq_top ?_

Goal state after:
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
K : ℝ≥0
f : X → Y
hf : AntilipschitzWith K f
s : Set Y
this✝³ : MeasurableSpace X := borel X
this✝² : BorelSpace X
this✝¹ : MeasurableSpace Y := borel Y
this✝ : BorelSpace Y
d : ℝ≥0
hd : μH[↑d] (f ⁻¹' s) = ∞
⊢ μH[↑d] s = ∞
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
⊢ (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet

Tactic:
by_cases hgf : (g ∘ₗ f).ker = ⊥

Goal state after:
case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
hgf : (g ∘ₗ f).ker = ⊥
⊢ (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet

case neg
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
hgf : ¬(g ∘ₗ f).ker = ⊥
⊢ (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
x : X
⊢ ‖kernel H x x‖ = ‖kerFun H x‖ ^ 2

Tactic:
by
  rw [sq, ← ContinuousLinearMap.norm_adjoint_comp_self, kernel_apply]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
s : Set X
hC : ∀ x ∈ s, ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
⊢ ∀ ε > 0, ∀ᶠ (n : ι) in l, ∀ x ∈ s, dist (f x) ((F n) x) < ε

Tactic:
intro ε hε

Goal state after:
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
s : Set X
hC : ∀ x ∈ s, ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (n : ι) in l, ∀ x ∈ s, dist (f x) ((F n) x) < ε
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : ℚ
⊢ innerProp' E ↑r

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : ℚ
x : E
⊢ ∀ (y : E), inner_ 𝕜 (↑r • x) y = (starRingEnd 𝕜) ↑r * inner_ 𝕜 x y
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : ℝ
⊢ innerProp' E ↑r

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : ℝ
x : E
⊢ ∀ (y : E), inner_ 𝕜 (↑r • x) y = (starRingEnd 𝕜) ↑r * inner_ 𝕜 x y
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
⊢ innerProp' E I

Tactic:
intro x

Goal state after:
case neg
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x : E
⊢ ∀ (y : E), inner_ 𝕜 (I • x) y = (starRingEnd 𝕜) I * inner_ 𝕜 x y
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : I * I = -1
h₁ : ‖-x - y‖ = ‖x + y‖
⊢ ‖-x + y‖ = ‖x - y‖

Tactic:
← neg_sub,

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : I * I = -1
h₁ : ‖-x - y‖ = ‖x + y‖
⊢ ‖-x + y‖ = ‖-(y - x)‖
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
⊢ innerProp' E r

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
x : E
⊢ ∀ (y : E), inner_ 𝕜 (r • x) y = (starRingEnd 𝕜) r * inner_ 𝕜 x y
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
d : ℝ
hd : 0 ≤ d
r : 𝕜
hr : r ≠ 0
s : Set E
this : ∀ {r : 𝕜} (s : Set E), μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH[d] s
⊢ μH[d] s ≤ ↑(‖r⁻¹‖₊ ^ d) * μH[d] (r • s)

Tactic:
refine Eq.trans_le ?_ (this (r • s))
      rw [inv_smul_smul₀ hr]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n ↦ 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), ediam (t n i) ≤ 1 / ↑n
n : ℕ
hn : 1 ≤ n
npos : 0 < ↑n
⊢ (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i

Tactic:
intro x

Goal state after:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n ↦ 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), ediam (t n i) ≤ 1 / ↑n
n : ℕ
hn : 1 ≤ n
npos : 0 < ↑n
x : ι → ℝ
⊢ (x ∈ univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) → x ∈ ⋃ i, t n i
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inr
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
v : E
s : Set ℝ
hv : v ≠ 0
⊢ μH[1] ((fun r ↦ r • v) '' s) = ‖v‖₊ • μH[1] s

Tactic:
have hn : ‖v‖ ≠ 0 := norm_ne_zero_iff.mpr hv

Goal state after:
case inr
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
v : E
s : Set ℝ
hv : v ≠ 0
hn : ‖v‖ ≠ 0
⊢ μH[1] ((fun r ↦ r • v) '' s) = ‖v‖₊ • μH[1] s
Module: Mathlib.Analysis.LocallyConvex.Montel

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹³ : NormedField 𝕜₁
inst✝¹² : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : IsTopologicalAddGroup E
inst✝⁷ : ContinuousSMul 𝕜₁ E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₂ F
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜₂ F
inst✝¹ : T1Space E
inst✝ : MontelSpace 𝕜₁ E
s : Set (E →SL_c[σ] F)
hs : IsOpen[UniformConvergenceCLM.instTopologicalSpace σ F {S | IsCompact S}] s
⊢ {S | IsCompact S} ⊆ {S | IsVonNBounded 𝕜₁ S}

Tactic:
intro x

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹³ : NormedField 𝕜₁
inst✝¹² : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : IsTopologicalAddGroup E
inst✝⁷ : ContinuousSMul 𝕜₁ E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₂ F
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜₂ F
inst✝¹ : T1Space E
inst✝ : MontelSpace 𝕜₁ E
s : Set (E →SL_c[σ] F)
hs : IsOpen[UniformConvergenceCLM.instTopologicalSpace σ F {S | IsCompact S}] s
x : Set E
⊢ x ∈ {S | IsCompact S} → x ∈ {S | IsVonNBounded 𝕜₁ S}
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
K : Submodule ℝ E := ℝ ∙ x
this✝ : finrank ℝ ↥K ≤ {x}.card
this : {x}.card = 1
⊢ 0 < finrank ℝ ↥Kᗮ

Tactic:
have : finrank ℝ K + finrank ℝ Kᗮ = finrank ℝ E := K.finrank_add_finrank_orthogonal

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
K : Submodule ℝ E := ℝ ∙ x
this✝¹ : finrank ℝ ↥K ≤ {x}.card
this✝ : {x}.card = 1
this : finrank ℝ ↥K + finrank ℝ ↥Kᗮ = finrank ℝ E
⊢ 0 < finrank ℝ ↥Kᗮ
Module: Mathlib.Analysis.LocallyConvex.WeakSpace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hs : Convex ℝ s
x : E
hx : (toWeakSpace 𝕜 E).toEquiv x ∈ (⇑(toWeakSpace 𝕜 E) '' closure s)ᶜ
this : ContinuousSMul ℝ E
f : StrongDual 𝕜 E
u : ℝ
hus : ∀ a ∈ closure s, RCLike.re (f a) < u
hux : u < RCLike.re (f x)
f' : StrongDual 𝕜 (WeakSpace 𝕜 E) :=
  let __LinearMap := ↑f ∘ₗ ↑(toWeakSpace 𝕜 E).symm;
  { toLinearMap := __LinearMap, cont := ⋯ }
hux' : u < (RCLike.reCLM ∘SL ContinuousLinearMap.restrictScalars ℝ f') ((toWeakSpace 𝕜 E) x)
hus' : closure (⇑(toWeakSpace 𝕜 E) '' s) ⊆ {y | (RCLike.reCLM ∘SL ContinuousLinearMap.restrictScalars ℝ f') y ≤ u}
⊢ (toWeakSpace 𝕜 E).toEquiv x ∈ (closure (⇑(toWeakSpace 𝕜 E) '' s))ᶜ

Tactic:
exact (hux'.not_ge <| hus' ·)

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] fun f ↦ g.comp f

Tactic:
refine continuous_of_dual_apply_continuous fun x z ↦ ?_
  have σ_bij : Function.Bijective σ₂₃ := ⟨σ₂₃.injective, RingHomSurjective.is_surjective⟩
  let σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
  let invPair : RingHomInvPair σ₂₃ σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
  let invPair_symm := invPair.symm
  let σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
    { toLinearEquiv := .ofBijective σ₂₃.toSemilinearMap σ_bij
      norm_map' _ := RingHomIsometric.norm_map }
  conv => enter [1, a]; rw [← σ_li.apply_symm_apply (z _), comp_apply, ← toCLM_apply]
  apply σ_li.continuous.comp
  exact continuous_dual_apply x <| σ_li.symm.toLinearIsometry.toContinuousLinearMap.comp <|
    z.comp g.toCLM

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
σ : ℝ
f : ℂ → E
x : ℝ
hx : 0 < x
⊢ |(2 * π)⁻¹| • ∫ (y : ℝ), ↑x ^ (-(↑σ + ↑y * I)) • f (↑σ + ↑y * I) =
    ↑x ^ (-↑σ) • ∫ (y : ℝ), cexp (2 * ↑π * (↑y * -↑(Real.log x)) * I) • f (↑σ + 2 * ↑π * ↑y * I)

Tactic:
have hx0 : (x : ℂ) ≠ 0 := ofReal_ne_zero.mpr (ne_of_gt hx)

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
σ : ℝ
f : ℂ → E
x : ℝ
hx : 0 < x
hx0 : ↑x ≠ 0
⊢ |(2 * π)⁻¹| • ∫ (y : ℝ), ↑x ^ (-(↑σ + ↑y * I)) • f (↑σ + ↑y * I) =
    ↑x ^ (-↑σ) • ∫ (y : ℝ), cexp (2 * ↑π * (↑y * -↑(Real.log x)) * I) • f (↑σ + 2 * ↑π * ↑y * I)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
this : EqOn (fun t ↦ ↑t ^ (s - 1) • f (a * t)) (fun t ↦ ↑a ^ (1 - s) • (fun u ↦ ↑u ^ (s - 1) • f u) (a * t)) (Ioi 0)
⊢ ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f (a * t) = ↑a ^ (-s) • ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t

Tactic:
rw [setIntegral_congr_fun measurableSet_Ioi this, integral_smul,
    integral_comp_mul_left_Ioi (fun u ↦ (u : ℂ) ^ (s - 1) • f u) _ ha,
    mul_zero, ← Complex.coe_smul, ← mul_smul, sub_eq_add_neg,
    cpow_add _ _ (ofReal_ne_zero.mpr ha.ne'), cpow_one, ofReal_inv,
    mul_assoc, mul_comm, inv_mul_cancel_right₀ (ofReal_ne_zero.mpr ha.ne')]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
⊢ ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x

Tactic:
intro x

Goal state after:
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
⊢ x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
⊢ ‖log (1 + z)‖ ≤ ‖z‖ ^ 2 * (1 - ‖z‖)⁻¹ / 2 + ‖z‖

Tactic:
by
  rw [← sub_add_cancel (log (1 + z)) z]
  exact norm_add_le_of_le (Complex.norm_log_one_add_sub_self_le hz) le_rfl

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
⊢ ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x

Tactic:
intro x

Goal state after:
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
⊢ x ∈ uIcc 0 (π / 2) →
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case refine_1
z : ℂ
hz : ‖z‖ < 1
⊢ Summable fun i ↦ ‖(-1) ^ (i + 1) * z ^ i / ↑i‖

Tactic:
refine (summable_geometric_of_norm_lt_one hz).norm.of_nonneg_of_le (fun _ ↦ norm_nonneg _) ?_

Goal state after:
case refine_1
z : ℂ
hz : ‖z‖ < 1
⊢ ∀ (b : ℕ), ‖(-1) ^ (b + 1) * z ^ b / ↑b‖ ≤ ‖z ^ b‖
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case hg
u : ℂ
hu : 0 < u.re
v : ℂ
⊢ ∀ x ∈ uIcc 0 (1 / 2), ContinuousAt (fun x ↦ (1 - ↑x) ^ (v - 1)) x

Tactic:
intro x

Goal state after:
case hg
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
⊢ x ∈ uIcc 0 (1 / 2) → ContinuousAt (fun x ↦ (1 - ↑x) ^ (v - 1)) x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in 0 / a..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
    a • ∫ (x : ℝ) in 0 / a..a / a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1))

Tactic:
← intervalIntegral.integral_comp_div _ ha.ne',

Goal state after:
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in 0 / a..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
    ∫ (x : ℝ) in 0..a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
⊢ 0 < (s + t).re

Tactic:
rw [add_re]

Goal state after:
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
⊢ 0 < s.re + t.re
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
n : ℕ
⊢ f x ≤ f 1 + x * log (↑n + 1) - x * log ↑n + logGammaSeq x n

Tactic:
rw [logGammaSeq, ← add_sub_assoc, le_sub_iff_add_le, ← f_add_nat_eq (@hf_feq) hx, add_comm x]

Goal state after:
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
n : ℕ
⊢ f (↑(n + 1) + x) ≤ f 1 + x * log (↑n + 1) - x * log ↑n + (x * log ↑n + log ↑n !)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
⊢ ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x

Tactic:
intro x

Goal state after:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
⊢ x ∈ Ioo 0 1 → HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm :
  Tendsto
    ((fun x_1 ↦ logGammaSeq x (x_1 - 1) + x * (log (↑(x_1 - 1) + 1) - log ↑(x_1 - 1)) - log (x - 1) + log (x - 1)) ∘
      fun a ↦ a + 1)
    atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
⊢ ((fun x_1 ↦
        (fun n ↦ logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b ↦ log (x - 1)) x_1) ∘
      fun a ↦ a + 1) =
    fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)

Tactic:
ext1 n

Goal state after:
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm :
  Tendsto
    ((fun x_1 ↦ logGammaSeq x (x_1 - 1) + x * (log (↑(x_1 - 1) + 1) - log ↑(x_1 - 1)) - log (x - 1) + log (x - 1)) ∘
      fun a ↦ a + 1)
    atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
n : ℕ
⊢ ((fun x_1 ↦
          (fun n ↦ logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
            (fun b ↦ log (x - 1)) x_1) ∘
        fun a ↦ a + 1)
      n =
    logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
c_zero_pos : 0 < cos 0
⊢ 0 ∈ closure (Ioo 0 (π / 2))

Tactic:
closure_Ioo pi_div_two_pos.ne,

Goal state after:
f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
c_zero_pos : 0 < cos 0
⊢ 0 ∈ Icc 0 (π / 2)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: a s : ℂ
hs : 0 < s.re + a.re
⊢ HasMellin ((Ioc 0 1).indicator fun t ↦ ↑t ^ a) s (1 / (s + a))

Tactic:
have := hasMellin_one_Ioc (by rwa [add_re] : 0 < (s + a).re)

Goal state after:
a s : ℂ
hs : 0 < s.re + a.re
this : HasMellin ((Ioc 0 1).indicator fun x ↦ 1) (s + a) (1 / (s + a))
⊢ HasMellin ((Ioc 0 1).indicator fun t ↦ ↑t ^ a) s (1 / (s + a))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) (log ∘ f)
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = y * f y
hf_pos : ∀ {y : ℝ}, 0 < y → 0 < f y
hf_one : f 1 = 1
⊢ EqOn (log ∘ f) (log ∘ Γ) (Ioi 0)

Tactic:
intro x

Goal state after:
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) (log ∘ f)
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = y * f y
hf_pos : ∀ {y : ℝ}, 0 < y → 0 < f y
hf_one : f 1 = 1
x : ℝ
⊢ x ∈ Ioi 0 → (log ∘ f) x = (log ∘ Γ) x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A :
  Tendsto
    (fun n ↦
      (↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ((∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
          ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n))))
    atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x ↦ Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
⊢ (fun n ↦ (∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ n)) =
    fun n ↦ (∫ (x : ℝ) in 0..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ n)

Tactic:
ext1 n

Goal state after:
case e'_3
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A :
  Tendsto
    (fun n ↦
      (↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ((∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
          ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n))))
    atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x ↦ Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
n : ℕ
⊢ (∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ n) =
    (∫ (x : ℝ) in 0..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
x : ℝ
⊢ (fun n ↦ π * x * ∏ j ∈ Finset.range n, (1 - x ^ 2 / (↑j + 1) ^ 2)) =
    Complex.re ∘ fun n ↦ ↑π * ↑x * ∏ j ∈ Finset.range n, (1 - ↑x ^ 2 / (↑j + 1) ^ 2)

Tactic:
ext1 n

Goal state after:
case e'_3
x : ℝ
n : ℕ
⊢ π * x * ∏ j ∈ Finset.range n, (1 - x ^ 2 / (↑j + 1) ^ 2) =
    (Complex.re ∘ fun n ↦ ↑π * ↑x * ∏ j ∈ Finset.range n, (1 - ↑x ^ 2 / (↑j + 1) ^ 2)) n